How to show that all even perfect numbers are obtained via Mersenne primes?

A number $n$ is perfect if it's equal to the sum of its divisors (smaller than itself). A well known theorem by Euler states that every even perfect number is of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is prime (this is what is called a Mersenne prime).

How is this theorem proved? The converse (showing that every number of that form is perfect) is simple (the divisors of such a number are powers of 2 and powers of 2 times the Mersenne prime, and it's an easy sum to calculate) but I couldn't find a proof for the theorem itself.

Solution 1:

Many elementary number theory texts have the proof, e.g., Hardy & Wright's. Here is an online proof from Chris Caldwell's Prime Pages.